Unit 4 · Topic 4.4 · about 20 minutes

Setting Up a Test for a Population Mean or Population Mean Difference

Turn a question about a population mean or a paired mean difference into a t-test setup that names the test, the parameter, the hypotheses and the conditions.

Predict first

A juice company fills bottles labeled 500 mL. A consumer group suspects the bottles are underfilled on average and plans to measure a random sample of them. Which pair of hypotheses fits the group's question?

Which test, for which parameter

When a question asks whether data give convincing evidence about a population mean, and the population standard deviation $σ$ is unknown, the test is the one-sample t-test for a population mean. In practice $σ$ is almost never known.

When the data come in matched pairs, such as the same people measured twice or each subject trying both treatments, find the difference within each pair and use the one-sample t-test for a population mean difference. The mechanics are the same; you just run them on one sample of differences.

Define the parameter in words before you write any symbols. The definition names the population, the response variable and the context: " $μ$ = the mean volume, in mL, of all bottles filled by the company." For paired data, add the order of subtraction: " $μ_{d}$ = the mean of (after minus before) resting heart rate, in beats per minute, for all members of the gym."

Hypotheses for one mean and for a mean difference from matched pairs
The question asks whether the mean is	One sample	Matched pairs
less than a value	$H_{0} : μ = μ_{0}$ , $H_{a} : μ < μ_{0}$	$H_{0} : μ_{d} = 0$ , $H_{a} : μ_{d} < 0$
greater than a value	$H_{0} : μ = μ_{0}$ , $H_{a} : μ > μ_{0}$	$H_{0} : μ_{d} = 0$ , $H_{a} : μ_{d} > 0$
different from a value	$H_{0} : μ = μ_{0}$ , $H_{a} : μ \neq μ_{0}$	$H_{0} : μ_{d} = 0$ , $H_{a} : μ_{d} \neq 0$

Here $μ_{0}$ is the value from the claim being tested, such as 500 mL. For matched pairs the null value is 0, which says the two treatments (or the before and after measurements) give the same results on average.

The direction of the alternative comes from the question, and you choose it before looking at the data. If the question says "changed" or "differs", use $\neq$ . A one-sided alternative needs a reason in the question, such as a suspicion of underfilling. With pairs, the direction also depends on your order of subtraction: if after minus before is negative when a program works, then $H_{a} : μ_{d} < 0$ .

The conditions

The conditions are the same three as for the interval in Topic 4.2:

Randomization condition: the data come from a random sample or a randomized experiment.
10% condition: when sampling without replacement, $n \leq 0.10 N$ .
Sample data condition: the population distribution is approximately normal, or $n \geq 30$ , or, if $n < 30$ , the sample data are free from strong skewness and outliers.

With matched pairs, all three are about the differences. The sample data condition asks whether there are at least 30 differences or, if there are fewer, whether a graph of the differences shows no strong skewness and no outliers. When the data come from a randomized experiment on volunteers rather than from a sample, the 10% condition does not apply, because nothing was sampled from a population.

Water time minus sports-drink time for 14 runners, in seconds

A positive difference means the runner was faster after the sports drink. The differences show no strong skew and no outliers.

Worked exampleSetting up a paired test

A coach randomly selects 14 of the more than 300 members of a large running club. Each runner sprints 200 meters on two different days, once after a sports drink and once after water, with the order decided by a coin flip. The dotplot above shows each runner's water time minus sports-drink time. Set up a test of whether 200-meter times are lower after the sports drink, on average.

Identify the procedure. Each runner gives two times, so the data are paired. Use a one-sample t-test for a population mean difference.
Define the parameter. $μ_{d}$ = the mean of (water time minus sports-drink time), in seconds, for all members of the club. With this order, a positive difference means faster after the drink.
State the hypotheses. $H_{0} : μ_{d} = 0$ , so the drink makes no difference on average, and $H_{a} : μ_{d} > 0$ , so times are lower after the drink on average.
Check conditions. Random: the runners were randomly selected and the order of the drinks was randomly assigned. 10%: 14 is less than 10% of the more than 300 members. Sample data: there are only 14 differences, and their dotplot shows no strong skewness and no outliers.

Answer.

A one-sample t-test for $μ_{d}$ with $H_{0} : μ_{d} = 0$ and $H_{a} : μ_{d} > 0$ , with all three conditions met. Carrying out the test is the job of Topic 4.5.

Check your understanding

Last year the mean commute time for employees of a large hospital was 31 minutes. A new parking garage has since opened, and the human resources office wants to know whether the mean commute time has changed. It will survey a random sample of employees. Which hypotheses are appropriate?

A dentist wants to know whether a new toothpaste leaves less plaque than her usual brand. Twenty randomly selected patients from her large practice each use both toothpastes for one month, in random order, and a plaque score is recorded at the end of each month. Which test is appropriate?

A gym measures the resting heart rate of 25 randomly selected members before and after a 12-week cardio program and computes (after minus before) for each member. Which is the correct parameter for a test of whether resting heart rate is lower after the program, on average?

A teacher randomly selects 12 of the 95 students in her school's robotics club and records how many minutes each one spent on the club's online forum last week. She wants to test whether the mean for all club members is more than 60 minutes. A dotplot of the 12 times is roughly symmetric with no outliers. Which condition is NOT met?

A study measures the typing speed (in words per minute) of 30 randomly selected students before and after a month of using a typing app, and records each difference as (before minus after). To test whether students type faster after the month, on average, the alternative hypothesis should be

Course alignment, for teachers

AP Statistics topic 4.4, Unit 4: Inference for Quantitative Data: Means.

Skill 2.C: Identify appropriate statistical inference methods.
Skill 2.E: Identify the null and alternative hypotheses.
Skill 4.E: Justify the use of a chosen statistical inference method by verifying conditions.